The Turn-Of-The-Month-Effect Evidence From Periodic Generalized Autoregressive Conditional...

7/24/2019 The Turn-Of-The-Month-Effect Evidence From Periodic Generalized Autoregressive Conditional Heteroskedasticity (P

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International Journal of Economic Sciences and Applied Research 7 (3): 43-61

The Turn-of-the-Month-Effect: Evidence from Periodic Generalized Autoregressive

Conditional Heteroskedasticity (PGARCH) Model

Eleftherios Giovanis1,2

Abstract

The current study examines the turn of the month effect on stock returns in 20 countries.

This will allow us to explore whether the seasonal patterns usually found in global data;

America, Australia, Europe and Asia. Ordinary Least Squares (OLS) is problematic as it leads

to unreliable estimations; because of the autocorrelation and Autoregressive Conditional

Heteroskedasticity (ARCH) effects existence. For this reason Generalized GARCH models are

estimated. Two approaches are followed. Thefirst is the symmetric Generalized ARCH (1,1)

model. However, previous studies found that volatility tends to increase more when the stock

market index decreases than when the stock market index increases by the same amount. In

addition there is higher seasonality in volatility rather on average returns. For this reason

the Periodic-GARCH (1,1) is estimated. Thefindings support the persistence of the specific

calendar effect in 19 out of 20 countries examined.

Keywords: Calendar Effects, GARCH, Periodic-GARCH, Stock Returns, Turn of the Month

Effect

JEL Classification:C22, G14

1. Introduction

Seasonal variations in production and sales of goods are a well known fact in

business and economics. Seasonality refers to regular and repetitive fluctuation in a time

series which occurs periodically over a span of less than a year. Similarly, stock returns

exhibits systematic patterns at certain times of the day, week or month. The existence of

seasonality in stock returns however violates an important hypothesis in finance that is

efficient market hypothesis.

Capital market efficiency has been a very popular topic for empirical research since

Fama (1970) introduced the theoretical analysis of market efficiency and proclaimed the

Efficient Market Hypotheses (EMH). Subsequently, a great deal of research was devoted to

1 University of Bologna, Department of Economics, Via Tanari Vecchia 9-2, 40121, Bologna, [email protected], [email protected] Royal Holloway University of London, Department of Economics


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Eleftherios Giovanis

investigating the randomness of stock price movements for the purpose of demonstrating

the efficiency of capital markets. Since then, all kinds of calendar anomalies in stock market

return have been documented extensively in the finance literature. The most common

calendar effects are the day of the week and the month of the year effect. However a curious

anomaly, the turn of the month effect, has been found, which has been firstly documentedby Ariel (1987). He examined the US stock returns and found that the mean return for stock

is positive only for days immediately before and during the first half of calendar months,

and indistinguishable from zero for days during the second half of the month.

The purpose of this paper is to investigate the turn of the month effect in stock

market indices around the globe and to test its pattern, which can be used for the optimum

asset allocation with result the maximization of profits. Because each stock market behaves

differently and presents different turn of the month effect patterns, the trading strategy

should be formed in this way where the buy and sell signals and actions will be varied in

each stock market index. Haugen and Jorion (1996) suggested that calendar effects shouldnot be long lasting, as market participants can learn from past experience. Hence, if the

turn of the month effect exists, trading based on exploiting this calendar anomaly pattern

of returns should yield extraordinary profits at least for a short time. Yet such trading

strategies affect the market in that further profits should not be possible: the calendar effect

should break down.

The majority of the studies examining the turn of the month effect use as main

tools statistical, from parametric and non parametric, test hypotheses to conventional

econometric approaches and regression models, as ordinary least squares and symmetric

GARCH estimations. To my knowledge this is thefirst study where the Periodic Generalized

Autoregressive Conditional Heteroskedasticity (PGARCH) model for the turn of the month

effect is employed.

The remainder of the paper has as follows: Section 2 discusses the literature review;

in section 3 the methodology is described and section 4 presents the data sample and

reports the summary statistics. Section 5 reports the results, while section six discusses the

concluding remarks.

2. Literature Review

Many researchers studied the turn of the month effect. One of the first studies is by

Ariel (1987), who obtained daily data for Center for Research in Security Prices (CRSP)

value-weighted and equally-weighted stock index returns from 1963 through 1981. Ariel

(1987), using descriptive statistics, finds that there are positive returns for the period starting

on the last trading day of the previous month through the first half of the next month,

followed by negative returns after the mid-point of the month. Also Ariel (1987) considers

the January effect and he finds that for both indexes the means of both the first and the

last nine trading days are lower when January is excluded from the analysis. Cadsby andRatner (1992) examined stock market indices in ten countries-CRSP value-weighted and

equally-weighted stock index returns for USA, Toronto stock exchange equally-weighted


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The Turn-of-the-Month-Effect: Evidence from Periodic Generalized AutoregressiveConditional Heteroskedasticity (PGARCH) Model

for Canada, Nikkei index for Japan, Hang Seng for Hong Kong, Financial times 500 share

or UK, All ordinaries index for Australia, Banca Commerciale index for Italy, Swiss Bank

Corporation Industrial index for Switzerland, the Commerzbank index for west Germany

and the Compagnie des Agents de Change General Index for France. The dates vary in

each index covering the period 1962-1989. Cadsby and Ratner (1992) define the turn-of-the-month effect as the last and the first three trading days of each month. Daily returns

are regressed on a constant and on a dummy variable, which equals at one for the turn-

of-the month days and zero for the other days, using ordinary least squares approach.

The coefficient of the dummy variable is statistically higher than zero at 1% level for

both value-weighted and equally-weighted stock indices of U.S.A. Also they reject the

null hypothesis for Canada, Switzerland and West Germany at the same significance level.

The same coefficient is statistically higher than zero at 5% level for United Kingdom and

Australia. However Cadsby and Ratner (1992) accept the null hypothesis for Japan, Hong

Kong, Italy and France.Jaffe and Westerfield (1989) obtain daily returns of stock market indices for four

countries. The specific indices and the periods they examine are Financial Times Ordinary

Share Index from January 2, 1950 to November 30, 1983 for UK; Nikkei Dow from January

5, 1970 to April 30, 1983 for Japan; Toronto Stock Exchange Index from January 2, 1977 to

November 30, 1953 for Canada; and Statex-Actuaries Index from January 1, 1973 to April

30, 1985 for Australia. They apply t-statisticsto test whether there is significant difference

between the intervals [-9, -2] and [-1, +9], where +1 denotes the first trading day of each

month and -1 denotes the last trading day of each month. The results are mixed as authors

find that there are higher returns of the first half of the month than returns of the last half of

the month for Canada, Australia and United Kingdom. Jaffe and Westerfield (1989) change

the intervals to [-10, -2] and [-1, +8] and they found positive significant returns only for

Australia, while positive returns are observed for Canada and United Kingdom; however

are statistically insignificant. The mean returns in the second half of the month are higher

than the first half for Japan and are significant at 1% level, suggesting a reverse monthly

effect. Finally, Jaffe and Westerfield (1989) estimated a model using as dependent variable

the daily returns of stock indices and independent variable a dummy, which takes value

one during the first trading days and the last trading day of each month and zero otherwise.

The coefficient of dummy variable is significant and positive for Canada, Australia and

United Kingdom, while is significant and negative for Japan. Ziemba (1991) examines

daily returns for NSA Japan during 1949-1988 for the intervals [-5, +2] and [-5, +7] and

applying descriptive and t-statisticsfinds that in these intervals returns are higher than any

other period. Finally, when the January effect is considered for the turn-of-the year effects,

this effect starts on day -7 and it has positive returns on every trading day until day +14.

McConnell and Xu (2008) examine the turn-of-the month effect for (CRSP) value-

weighted and equally-weighted stock index returns in USA obtaining daily data during

period 1926-2005. In addition, they examine the same effect using two sub-periods, 1926-1986 and 1987-2005. Also they test the turn-of-the month effect for other 34 countries

McConnell and Xu (2008) define the turn-of-the month interval as [-1, +3] and they found


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that the specific calendar effect exists for USA and for other 30 out of 34 countries except

Argentina, Colombia, Italy, and Malaysia. The methodology they follow is descriptive

and they use t-statisticsto test whether the mean accumulative returns in the turn-of-the

month interval are significant positive and different from zero and higher than the mean

cumulative returns in the rest days of each trading month.Martikainen et al. (1995) used daily returns of Finnish Options Index from May 2,

1988 to October 14, 1993. They examined the interval [-1, +4] as the turn-of-the month and

they apply t-statisticsto test if the mean returns of this interval are positive and significant

different from zero. Martikainen et al. (1995) found that these positive and significant

returns are observed in the interval [-5, +5]. Kunkel et al. (2003) used daily closing prices

for 19 countries from August 1, 1988 to July 31, 2000 to examine the turn-of-the month

effect , which is defined as the interval [-1,+3]. Kunkel et al. (2003) regress daily returns on

18 dummy variables, which for example dummy D-9takes value one if returns correspond

to trading day -9 , continuing through D9which corresponds to trading day 9. The methodwhich is applied is ordinary least squares. Over the 4-day turn-of-the month interval, all

countries have at least one positive and statistically different from zero return, while most

of them have two to four positive and statistically different from zero returns. Six countries

have negative returns during this 4-day turn-of-the month period; however none of these

returns are statistically insignificant. Finally Kunkel et al. (2003) regressed daily returns

on a constant and on a dummy, where the latter takes value one of returns are corresponding

in the turn-of-the month effect [-1, +3] interval and zero otherwise. The coefficients of this

regression shows that there are positive mean returns in every country during the [-1, +3]

interval.

Nikkinen et al. (2007) used daily data of SP100 stock market and VIX volatility

indices data from January 1995 to December 2003. In the study by Nikkinen et al. (2007)

daily returns of SP100 are regressed on two dummies. The first dummy takes value one if

returns refer on the interval [-9, +9] and zero otherwise, while the second variable takes

value one if returns refer on the remained days of the month and zero otherwise. Nikkinen

et al. (2007) find that the turn-of-the month effect is strongest in the [+1 +3] interval.

Aggarwal and Tandon (1994) obtained daily data for 18 countries. The turn-of-the month

is defined as the interval [-4, +4]. Aggarwal and Tandon (1994) used t-statisticsand they

found that there are significantly higher returns in the [-1, +3] interval in ten countries.

Lakonishok and Smidt (1988) used ninety year daily data of Dow Jones Industrial

Average from January 4, 1897 through June 11, 1986. Lakonishok and Smidt (1988) use

t-statistics to test the difference in the average returns between turn-of-the-month interval

and non turn-of-the-month and they find that the turn-of-the month effect strongly exists in

the [-1, +3] interval. Marquering, et al. (2006) used daily and monthly data of Dow Jones

Industrial Average (DJIA) during period 1960-2003, with two sub-periods of estimation;

1960-1981 and 1982-2003. Marquering, et al. (2006) found that the turn-of-the-month

effect still exists, while the other calendar effects, including the day of the week and themonth of the year effect, disappear. Tonchev and Kim (2004) used daily values PX-50 and

PX-D Indices of Czech Republic, the SAX Index for Slovakia and the SBI-20 and SBI-


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20NT indices for Slovenia. The periods are 1 January 1999- 18 June 2003 for the Czech

Republic and 4 July 2000- 18 June 2003 for Slovakia. Tonchev and Kim (2004) studied the

day-of-the week, January, turn-of-the month, the half month and holiday effects. Tonchev

and Kim (2004) regressed the daily returns on six dummies where dummy D-3is equal with

one if returns correspond to the trading day 3, continuing through up to D3, which is equalwith one if returns correspond to the trading day 3. All models are estimated with OLS and

GARCH(1,1). Tonchev and Kim (2004) found that the turn-of-the-month effect does not

exist. Giovanis (2009) examined the turn of the month effect in 55 stock market indices

using bootstrapping t-statistics, concluding that the turn of the month effects is present

in 36 indices. Georgantopoulos and Tsamis (2014) examined various calendar anomalies,

including the turn-of-the-month effect, in stock returns on Istanbul Stock Exchange (ISE)

over an eight years period (4/1/2000 4/1/2008) by Ordinary Least Squares (OLS) and

Generalized Autoregressive Conditional Heteroskedasticity (GARCH 1,1) models. Testing

the presence of the turn of the month effect the authors found that this market anomaly isstrongly present in the ISE.

Hansen and Lunde (2003)derived a test for calendar anomalies, which controls for

the full space of possible calendar effects. The countries examined are: Denmark, France,

Germany, Hong Kong, Italy, Japan, Norway, Sweden, Japan, UK, and USA. The authors

investigated various calendar effects including the turn-of-the-month effect and they found

significant calendar effects in most series examined. However, in recent years it seems that

the calendar effects have diminished, while most robust significance is found for small-cap

stock indices, where calendar effects are generally found to be significant, across countries

and subsamples. Zwergel (2014) examined the turn of the month effect using the indices;

Germany (DAX), Japan (Nikkei 225), UK (FTSE 100) and US (S&P 500) during the period

January 1991 and November 2005. Zwergel (2014) argues that the turn of the month effect

seems to be exploitable by using a futures trading strategy, even after transaction costs

and slippage deductions due to the fact that turn of the month effect is quite volatile and

that the liquidity at the close trades are assumed to be executed, is too low for institutional

investors. Thus, the investors they would probably be paying higher prices than the closing

prices when opening a long position and receiving lower prices when closing the position.

Sharma and Narayan (2014) examined whether the turn-of-the-month affects firm returns

and firm return volatility differently depending on their sector and size. Using 560 firms

listed on the NYSE Sharma and Narayan (2014) found evidence that the turn-of-the-month

affects returns and return volatility of firms. However, these effects depend on firm location

and size.

On the other hand, other studies examine additional factors having impact on stock

returns. A study by Vazakidis and Athianos (2010) examines the reaction of the Athens

Stock Exchange (ASE) to dividend announcements by a sample of firms listed at the FTSE/

ATHEX 20 and FTSE/ATHEX Mid 40 for a fixed period 2004-2008, before and after the

day of the announcement (event day). The authors test the hypotheses that there is nosignificant abnormal activity by the stock prices during the examined period and thus, the

irrelevance theory introduced by Miller and Modigliani (1961) stands true. Using various


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event windows, no longer than 20 days, Vazakidis and Athianos (2010) reject the irrelevance

theory and the hypothesis of no abnormal stock returns. alet al. (2011) examined the

volatility shifts and persistence in variance using data for the sector indices of Istanbul

Stock Exchange market. The authors extended the exponential generalized autoregressive

conditional heteroskedasticity (EGARCH) model, proposed by Nelson (1991), by takingaccount of the volatility shifts which are determined by using iterated cumulative sums

of squares (ICSS) and modified ICSS algorithms such as Kappa-1 (-1) and Kappa-2 (-

2). Their findings support that the inclusion of volatility shifts in the model substantially

reduces volatility persistence and suggest that the sudden shifts in volatility should not be

ignored in modelling volatility for Turkish sector indices.

Sariannidis (2010) using Generalized Autoregressive Conditional Heteroskedasticity

(GARCH) models examined the effects of capital and energy markets returns and exchange

rate of the U.S. Dollar/ Yen on sugar features. More specifically, Sariannidis (2010)

examines crude oil, Ethanol, SP500 and exchange rate of the U.S. Dollar/ Yen and hefound that the higher energy prices, Crude oil and Ethanol, positively influence the sugar

market, while the effects of U.S. Dollar/ Yen are negative on sugar market. Therefore, this

study is suggested for future research as the calendar effects can be influenced by additional

macroeconomic factors.

In recent years, there has been considerable interest in the autoregressive conditional

heteroskedasticity (ARCH) disturbance model introduced by Engle (1982). Since their

introduction, the ARCH model and its various generalizations, especially the generalized

ARCH (GARCH) model introduced by Bollerslev (1986), have been particularly popular

and useful in modelling the disturbance behaviour of the regression models of monetary

and financial variables. Srinivasan and Ibrahim (2012) used a bivariate Error Correction

Model Exponential GARCH (ECM-EGARCH) to examine the news effects from the spot

exchange rates market to the volatility behaviour of futures market. Sariannidis et al. (2009)

used the GARCH model to examine the relationship between Dow Jones Sustainability

Index World (DJSI.-World) returns to 10 year bond returns and Yen/U.S. dollar exchange

rate. An extensive survey of the theory and applications of these models is given by

Bollerslev et al. (1992).

Previous studies found that seasonality in financial-market volatility is pervasive.

Gallant et al. (1992) reported that the historical variance of the Standard and Poors

composite stock-price index in October is almost ten times the variance for March. Similarly,

Bollerslev and Hodrick (1999) found evidence for significant seasonal patterns in the

conditional heteroskedasticity of monthly stock-market dividend yields. Regarding daily

frequency studies demonstrated that daily stock-return and foreign-exchange-rate volatility

tend to be higher following non-trading days, although proportionally less than during the

time period of the market closure (French and Roll, 1986; Baillie and Bollerslev, 1989).

At the intraday level, Wood et al. (1985) found that the variances of stock returns over the

course of the trading day present a U-shaped pattern. Similar patterns in the volatility ofintraday foreign-exchange rates are reported in other studies (Baillie and Bollerslev, 1991;

Harvey and Huang, 1991; Dacorogna et al., 1993).


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3. Data and Methodology

3.1 Data and Summary Statistics

The data are daily closed prices of stock market indices. The analysis is conducted interms of daily returns which is defined as r = log(Pt/Pt-1). More specifically, in Table 1 we

present the countries and the indices symbols. The final period is 31 December 2013 for all

series except from the starting period, where it is shown in Table 1.

In Table 2 the descriptive statistics for stock market indices returns in 10 countries

are reported. In all cases mean returns are very low and in some countries are negative as

in Italy and Taiwan. As it was expected leptokurtosis is observed in all stock returns, as the

value of kurtosis is very high reaching even 99 in the case of Australia.

Heavy tails are commonly found in daily return distributions. Negative skewness is

presented in all series expect from Brazil, Greece, Malaysia, and Mexico. In addition basedon Jarque-Bera statistic and the probability it is concluded that the normality assumption

in the time series examined is rejected, supporting the non-normal distribution of the stock

index returns examined in this study. One can use median return instead of mean to represent

returns. Based on median returns, Argentina and Brazil report the highest return followed

by India and Indonesia. On the other hand, the lowest median returns are presented in

Greece, followed by China and Taiwan.

Table 1: Stock Market Indices and estimating periods

Countries Period Countries Period

Argentina (MERVAL

INDEX)

9 October 1996 Indonesia (JKSE

Composite Index)

2 July 1997

Australia (All

ordinaries Index)

9 January 2001 Italy (MIBTEL INDEX) 2 January 1998

Austria (ATX INDEX) 12 November 1992 Japan (Nikkei 225) 5 January 1984

Brazil (IBOVESPA

INDEX)

28 April 1993 Malaysia (KLSE

INDEX)

6 December 1993

China (Shanghaicomposite Index)

4 July 1997 Mexico (IPC INDEX) 11 November 1991

France (CAC 40

INDEX)

2 March 1990 Netherlands (AEX

INDEX)

13 October 1990

Germany

(DAX INDEX)

27 November 1990 Singapore (STI INDEX)

Greece (GENERAL

INDEX)www.enet.gr

5 January 1998

Taiwan (TSEC weighted

index)

3 July 1997

Hong Kong (HANGSENG INDEX) 2 January 1987 UK (FTSE-100) 3 April 1984

India (BSE SENSEX) 2 January 1997 USA (S&P 500) 4 January 1950


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Table2:Descriptivestatisticsforstockreturnsin10countries

ARGENTINA

AUSTRAL

IA

AUSTRIA

BRAZIL

CHINA

FRANCE

GERMANY

GREECE

HONG

KONG

INDIA

Mean

0.000521

0.000271

0.000232

0.001495

0.000114

0.000141

0.000323

0.000247

0.000328

0.000390

Median

0.001011

0.000514

0.000641

0.001433

0.000110

0.000333

0.000786

0.000013

0.000529

0.000977

Maximu

m

0.161165

0.060666

0.120210

0.288325

0.094008

0.105946

0.107975

0.240000

0.172470

0.159900

Minimu

m

-0.147649

-0.28713

4

-0.102526

-0.172082

-0.092562

-0.094715

-0.098709

-0.24000

-0.405420

-0.118092

St.deviation

0.021687

0.010024

0.013730

0.023977

0.015081

0.014176

0.014444

0.018745

0.017398

0.016486

Skewne

ss

-0.284545

-3.78888

8

-0.387938

0.490767

-0.117682

-0.028677

-0.106106

0.108420

-2.386760

-0.090520

Kurtosis

8.356421

99.66951

10.57300

12.85889

7.937991

7.492765

7.745093

35.55614

59.90492

8.552297

Jarque-B

era

5,130.839

291,553.8

12,638.12

20,949.22

4,357.296

5,078.181

5,498.291

118096.0

912,660.0

5,248.904

Probability

0.000000

0.000000

0.000000

0.000000

0.000000

0.000000

0.000000

0.000000

0.000000

0.000000

INDONESIA

ITALY

JAPAN

MALAYSIA

MEXICO

NETHER-

LANDS

SINGAP

ORE

TAIWAN

UK-FTSE

100

USS&P500

Mean

0.000440

-6.16E-0

5

6.71E-05

0.000121

0.000614

0.000213

0.000209

-1.16E-05

0.000242

0.000292

Median

0.000926

0.000456

0.000391

0.000277

0.000739

0.000690

0.000229

0.000152

0.000212

0.000468

Maximu

m

0.131278

0.108742

0.132346

0.208174

0.121536

0.100283

0.128738

0.085198

0.093891

0.109572

Minimu

m

-0.127318

-0.08599

1

-0.161375

-0.241534

-0.143145

-0.095903

-0.105446

-0.099360

-0.130221

-0.228997

St.deviation

0.017278

0.015678

0.014615

0.014583

0.015652

0.013993

0.012615

0.015307

0.011015

0.009758

Skewne

ss

-0.188689

-0.07087

4

-0.297679

0.408046

0.020908

-0.147454

-0.086594

-0.150647

-0.49502

-1.030113

Kurtosis

9.854901

7.084830

11.14954

52.07823

8.765888

9.340613

11.45198

5.666394

12.88787

30.69292

Jarque-B

era

7,876.956

2,846.968

20,528.80

497,729.2

7,682.877

9,078.759

19,441

.74

1,220.773

32,192.60

517,372.2

Probability

0.000000

0.000000

0.000000

0.000000

0.000000

0.000000

0.000000

0.000000

0.000000

0.000000


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3.2 Stationarity and Unit Root Tests

In this section ADF test statistic (Dickey and Fuller, 1979) is applied in order to

examine whether the stock returns examined in this study stock returns are stationary as it

was expected. The ADF test can be defined by testing the following equation:

Rt= + t + Rt-1+ lags of Rt+ t (1)

and the hypotheses we test are:

H0: =1, =0 => Rt~ (0) with drift

against the alternative

H1: || Rt~ (1) with deterministic time trend

In Table 3 the results of ADF test are reported. Based on thet-statistics, the stock

returns are stationary. The stationarity is supported also based on additional tests, such as

the Dickey-Fuller (DF), the Phillips-Perron (PP) and the KwiatkowskiPhillipsSchmidt

Shin (KPSS) test.

Table 3: ADF test for stock returns in 20 countries

Countries Test ADF t-statistic Countries Test ADF t-statistic

ARGENTINA -51.362 INDONESIA -44.902

AUSTRALIA -36.224 ITALY -48.408

AUSTRIA -59.857 JAPAN -58.606

BRAZIL -58.165 MALAYSIA -26.618

CHINA -50.483 MEXICO -45.397

FRANCE -68.537 NETHERLANDS -39.251

GERMANY -67.875 SINGAPORE -44.335

GREECE -27.667 TAIWAN -50.785

HONG KONG -39.636 UK-FTSE 100 -39.451

INDIA -49.398 US S&P 500 -86.604

* MacKinnon critical values for rejection of hypothesis of a unit root at 1%, 5% and 10% are

-3.4786, -2.8824 and -2.5778 respectively.


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3.3 Symmetric GARCH Model

The consequences of heteroskedasticity are problematic in general, and it is well

known that the consequences of heteroskedasticity for OLS estimation are very serious.

Although parameter estimates remain unbiased, they are no longer efficient, meaning theyare no longer best linear unbiased estimators (BLUE) among the class of all the linear

unbiased estimators. For this reason GARCH and PGARCH models are employed in this

study to account for autocorrelation, heteroskedasticity and volatility clustering.

The turn-of-the month (TOM) effect is defined as the interval [-1, +3], where -1

is the last trading day of each month and continuing until +3, which is the third trading

day of each month. The general form of a GARCH (p,q) model, which was proposed by

Bollerslev (1986) is:

2 2 21 1

1 1

[ | ]

p q

t t t i t j t j

i j

E a

(2)

The GARCH (1,1) model will be:

2

0 1 , ~ (0, )t N t t R D D (3)

2 2 2

1 1 2 1t t ta u a (4)

where (3) and (4) indicate the mean and the variance equations respectively.Rtdenotes the

daily stock returns,DTOMis a dummy variable obtaining value 1 for mean returns belongingin the TOMinterval [-1, +3] and 0 otherwise, DNTOMis a dummy variable obtaining value

1 for mean returns not belonging in the NTOM interval and 0 otherwise and t is the

disturbance term. Based on the turn of the month effect, it is expected that the coefficient0

will be significant positive and higher than coefficient 1. Alternatively, it is expected that

coefficient 1 will be insignificant or negative.

Regarding the diagnostic tests, firstly ARCH effects are tested using the Breusch-

Pagan Lagrange Multiplier (LM) test. To test for ARCH of order p the following auxiliary

regression model is considered:

2 2 2 2

0 1 1 2 2 .......t t t p t p t (5)

Under the null hypothesis of no ARCH,

0 1 2..... 0pH (6)

The hypothesis can be tested using the familiar statistic:

2 2 ( )T R x p (7)

The second diagnostic test is the autocorrelation test on residuals. The Ljung-Box-

Pierce Q-statistic (Box and Pierce, 1970; Ljung and Box, 1978) is applied.


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Let^

(1)e ,.........,^

( )e n be the standardized residuals from fitting a time series regression

model, and let (34) be their autocorrelations.

^ ^

^1

^2

1

( ) ( )( ) , 1, 2,....

( )

n

t k

n

t k

e t e t k r k for k n

e t

(8)

If the model is correct, the Ljung-Box-Pierce Q-statistic is:

2^ ^1

1

( ) ( 2) ( ) ( )m

k

Q r n n n k r k

(9)

where (9) is asymptotically distributed asx2with m pdegrees of freedom wherepdenotes

the number of parameters in the model. The null hypothesis is that there is no autocorrelation

in the residuals. Various lags have been used; however for the ARCH LM test 5 lags have

been used and for the autocorrelation test 12 lags.

3.4 Periodic GARCH Model

In this section the methodology of Periodic GARCH (1,1) is provided, which have

been proposed by Bollerslev and Chysels (1996). The class of P-GARCH processes maybe defined as:

~

1[ | ] 0s

t tE

(10)

wheres(t)refers to the stage of the periodic cycle at time t. The general form of Periodic-

GARCH model is:

2 2 2~ ~ ~ ~

11 ( ) ( ) ( )

1 1

[ | ]q p

st t t t jt s t is t js t

i j

E a

(11)

In this study the Periodic GARCH (1,1) used for the turn-of-the month effect and

regression (2) is the following:

2 2 2 2 2

1 1 2 1 1 1 2 1t t t st s st s t st s t a u a d d a u d a (12)

In the variance equation (12) the coefficients definition remain the same as in (4),

with the difference that dstequals with one if sis the stage of the periodic cycle at time

tand dst

=0otherwise. More precisely stage of the periodic cyclesis equal at 1 for days

belonging in the TOMinterval and 0 otherwise.


12/19

54


4. Empirical Results

In Tables 4 and 5 the GARCH(1,1) and PGARCH(1,1) estimates respectively are

reported. Based on these results the coefficient0is always positive, significant and higher

than coefficient 1with the exception of Australia. Therefore the main conclusion is thatthe turn of the month effect is presented in 19 out of 20 stock market indices examined. The

findings are consistent with other studies (Ariel, 1987; Cadsby and Ratner, 1992; Giovanis,

2009). The coefficient0is always significant at 1% level, while in some cases coefficient1

is statistically insignificant. It should be noticed that different samples have been employed.

More specifically, three samples have been used. Firstly, from the starting period of each

stock market index up to 2007 before the financial crisis. The second sample is the period

2008-2009 and the third sample is from the starting period for each stock market index

up to 2013. However, the results remain the same, only changing the magnitude of the

coefficients around 1-2 %, and they are not presented due to space limitations. Nevertheless,the conclusion change only for Australia, where the turn of the month effect exists, using

the first two samples, but not when the whole period is included in the analysis, obtaining

also the post-financial crisis period 2010-2013.

Regarding the diagnostic tests PGARCH outperforms the GARCH model based on

Log-Likelihood, on Akaike Information Criterion (AIC) and Schwarz Bayesian Criterion

(SBC). In addition, in most cases the GARCH model solves for autocorrelation and ARCH

effects. On the other hand, PGARCH solves for these problems in Mexico, as well as, in

Netherlands for autocorrelation at 10%. However, in some countries the problems still

remain. More specifically, in Hong Kong both models do not solve for the autocorrelation

and ARCH effects, in Netherlands for autocorrelation at 10% and for ARCH effects at 5%

and 10% and in UK for ARCH effects at 10% level.

Moreover, the condition that a1+a2


13/19

55


Table4:GARCH

(1,1)estimationofm

eanequation(3)andvarianceequation(4)

Coun

tries

0

1

1

2

AIC

SBC

LL

LBQ2(12)

ARCH-

LM(5)

ARGENTINA

0.0024

(3.824)*

0.0007

(2.701)*

1.46e-05

(9.892)*

0.114

(16.394)*

0.855

(95.238)*

-5.080

-5.070

10,807.157

8.974

[0.705]

1.006

[0.4122]

AUSTRALIA

0.00032

(6.420)*

0.00147

4

(2.621)*

5.57e-05

(10.929)*

0.215

(75.850)*

0.734

(74.604)*

-6.801

-6.795

20,796.31

7.011

[0.857]

1.083

[0.3669]

AUSTRIA

0.0019

(5.840)*

0.0005

(3.297)*

2.80e-06

(7.429)*

0.0987

(18.707)*

0.885

(136.14)*

-6.151

-6.145

16,100.75

11.161

[0.523]

0.845

[0.5168]

BRA

ZIL

0.0054

(6.723)*

0.00081

(1.991)**

7.30e-06

(7.138)*

0.117

(15.768)*

0.863

(92.089)*

-4.853

-4.845

12,881.77

13.863

[0.310]

1.690

[0.1333]

CHINA

0.0014

(2.670)*

-0.0001

(-0.556

)

6.31e-06

(11.400)*

0.0961

(18.760)*

0.845

(77.084)*

-5.635

-5.624

12,310.71

8.554

[0.710]

0.113

[0.9894]

FRA

NCE

0.00198

(4.890)*

0.0002

(0.984

)

3.02e-06

(7.994)*

0.0855

(12.603)*

0.898

(113.54)*

-6.022

-6.015

17,934.18

10.176

[0.601]

1.440

[0.2061]

GERM

ANY

0.0015

(4.094)*

0.0005

(3.328)**

3.36-06

(8.531)*

0.0833

(12.133)*

0.897

(104.09)*

-6.019

-6.012

17,462.67

2.187

[0.999]

0.303

[0.9107]

GRE

ECE

0.0025

(4.537)*

0.0001

(0.379

)

5.57e-06

(5.570)*

0.0154

(15.637)*

0.837

(82.562)*

-5.602

-5.591

15,496.013

12.103

[0.511]

1.573

[0.249]

HONG

KONG

0.0021

(5.586)*

0.0007

(3.711)*

6.73e-06

(16.738)*

0.1382

(42.535)*

0.845

(199.68)*

-6.694

-6.688

19,172.44

165.70

[0.000]

35.316

[0.000]

IND

IA

0.0028

(6.434)*

0.0006

(3.054)**

4.30e-06

(7.112)*

0.1067

(13.645)*

0.881

(78.324)*

-5.546

-5.536

11,522.97

11.043

[0.525]

0.345

[0.8856]


14/19

56


Table4:(cont.)GARCH

(1,1)estimationofmeanequation(3)andvarianceequation(4)

Cou

ntries

0

1

1

2

AIC

SBC

LL

LBQ2(12)

ARCH-LM

(5)

INDO

NESIA

0.0019

(3.909)*

0.00088

(4.015)*

6.01e-06

(9.970)*

0.1336

(15.485)*

0.854

(91.517)*

-5.457

-5.446

11,227.33

8.9700

[0.705]

0.327

[0.8966]

ITALY

0.0015

(3.478)*

0.00

02

(1.0

53)

1.47e-06

(5.048)*

0.1028

(11.839)*

0.888

(90.571)*

-6.437

-6.425

11,968.15

19.849

[0.070]

2.316

[0.0413]

JA

PAN

0.0014

(4.439)*

0.00

06

(4.734)*

2.52e-06

(10.191)*

0.1228

(35.319)*

0.874

(215.64)*

-5.977

-5.971

21,821.54

6.6564

[0.879]

0.666

[0.6489]

MAL

AYSIA

0.00099

(3.256)*

-9.62e-05

(-1.0

40)

3.75e-06

(14.708)*

0.1813

(19.549)*

0.823

(124.74)*

-6.226

-6.217

15,654.33

8.2017

[0.769]

0.457

[0.8083]

ME

XICO

0.0024

(6.055)*

0.00

05

(3.033)*

6.81e-06

(7.976)*

0.1020

(19.629)*

0.890

(113.116)*

-5.627

-5.620

15,996.79

24.964

[0.015]

3.083

[0.0088]

NETHE

RLANDS

0.0017

(5.501)*

0.00

03

(2.341)**

1.32e-06

(5.374)*

0.0985

(14.556)*

0.8963

(135.34)*

-6.254

-6.246

16,738.62

21.142

[0.048]

2.922

[0.0122]

SING

APORE

0.0014

(5.368)*

0.00

02

(1.675)***

3.95e-06

(16.287)*

0.1348

(15.177)*

0.844

(54.065)*

-5.736

-5.725

20,428.87

2.923

[0.988]

0.267

[0.9307]

TAIWAN

0.0026

(5.386)*

0.00

01

(0.4

98)

1.68e-06

(4.175)*

0.0854

(11.554)*

0.9092

(119.45)*

-5.575

-5.565

11,733.02

20.484

[0.058]

3.366

[0.0049]

UK-FTSE100

0.0016

(6.539)*

0.00

03

(3.169)*

1.67.e-06

(6.502)*

0.0881

(17.923)*

0.889

(133.65)*

-6.566

-6.560

25,481.57

12.153

[0.433]

2.158

[0.065]

USS&P500

0.00127

(8.744)*

0.00021

(2.961)*

7.15e-06

(11.161)*

0.0815

(50.132)*

0.9120

(416.11)*

-6.862

-6.859

54,787.79

13.239

[0.352]

1.333

[0.2462]

z-statisti

cbetweenbrackets,p-valuesbetweensquarebrackets,

*den

otessignificancein0.0

1leve

l,

**denotessignificancein0.0

5leveland

***denotessignificancein0.1

0levelAIC

andSBCrefertoAkaikeandSchwarzinformationcriteria,

LListheLogLikelih

ood,

LBQ2i

s

theLjun

g-Boxtestonsquaredstandardizedresiduals


15/19

57


Table5:Periodic-GARCH

(1,1)estimation

ofmeanequation(3)and

varianceequation(12)

Countr

ies

0

1

1

2

s

1s

2s

AIC

SBC

LL

LBQ

2

(12)

ARCH-LM

(5)

ARGENT

INA

0.0

025

(3.6

41)*

0.0

007

(2.6

79)*

1.5

3e-05

(8.832)*

0.1

282

(17.6

95)*

0.8

423

(73.7

67)*

0.4

739

(1.7

01)***

-0.0

892

(-4.8

72)*

0.0

962

(1.8

16)***

-5.0

95

-5.0

83

10,8

15.4

6

10.900

[0.53

7]

1.3

00

[0.2

607]

AUSTRA

LIA

0.0

006

(3.5

25)*

0.0

007

(7.1

79)*

5.37E-06

(7.307)*

0.1

844

(74.0

52)*

0.7

813

(90.1

16)*

-0.6

021

(-1.2

62)

-0.0

275

(-1.4

14)

0.0

579

(1.1

91)

-6.8

03

-6.7

94

20,8

03.3

9

6.94

5

[0.86

1]

1.0

80

[0.3

687]

AUSTR

IA

0.0

019

(5.7

92)*

0.0

005

(3.5

17)*

1.3

9e-05

(4.528)*

0.1

095

(17.3

39)*

0.8

849

(96.5

50)*

0.0

830

(2.4

92)**

-0.0

463

(-2.8

19)*

0.0

273

(0.6

74)

-6.1

53

-6.1

47

16,1

04.3

5

11.2

30

[0.50

9]

0.6

46

[0.6

644]

BRAZIL

0.0

039

(5.4

49)*

0.0

005

(2.0

63)**

3.99E-05

(3.396)*

0.1

073

(17.7

25)*

0.8

817

(97.4

34)*

0.1

586

(0.8

61)

-0.0

268

(-1.4

96)**

0.0

103

(0.2

82)

-4.8

61

-4.8

52

12,8

82.7

2

18.735

[0.11

5]

2.1

17

[0.0

941]

CHINA

0.0

016

(3.6

97)*

-0.0

002

(-0.2

05)

1.07E-05

(3.093)*

0.1

115

(17.3

64)*

0.8

467

(123.6

3)*

0.0

579

(6.2

23)*

0.0

433

(2.6

47)*

0.0

333

(1.4

14)

-5.6

40

-5.6

26

12,3

49.6

6

8.97

0

[0.70

5]

0.3

27

[0.8

966]

FRANCE

0.0

017

(4.8

03)*

0.0

002

(1.6

69)***

3.31E-06

(1.5

23)

0.0

940

(14.6

25)*

0.8

825

(101.8

1)*

-0.0

144

(-1.4

20)

-0.0

422

(-2.5

01)**

0.0

962

(2.1

93)**

-6.0

24

-6.0

13

17,9

33.2

1

7.34

4

[0.83

4]

1.1

68

[0.3

218]

GERMA

NY

0.0

016

(4.6

71)*

0.0

005

(3.3

40)*

3.52E-06

(5.8

59)*

0.0

778

(4.9

44)*

0.9

031

(33.6

36)*

0.0

928

(4.2

98)*

-0.0

281

(-1.5

24)

0.0

826

(2.0

87)**

-6.0

26

-6.0

15

17,4

64.6

2

3.03

1

[0.99

5]

0.4

93

[0.7

813]

GREECE

0.0

025

(2.9

54)*

-0.0

0010

(-0.2

20)

9.61E-05

(16.7

29)*

0.2

479

(13.9

20)*

0.4

579

(74.5

18)*

-0.0

497

(-0.0

31)

0.0

458

(1.6

45)***

0.0

261

(26.7

31)*

-5.6

12

-5.5

97

15,5

02.2

9

11.8

65

[0.59

2]

1.4

93

[0.3

16]

HONGKONG

0.0

022

(5.1

69)*

0.0

007

(3.3

34)*

1.63E-05

(10.8

03)*

0.1

027

(12.2

37)*

0.8

522

(71.3

40)*

0.0

644

(3.8

96)*

-0.0

381

(-2.5

00)**

0.1

413

(5.4

47)*

-5.7

03

-5.6

92

19,1

72.8

3

148.79

[0.00

0]

31.4

01

[0.0

00]

INDIA

0.0

029

(6.5

39)*

0.0

006

(3.0

55)*

2.99e-06

(9.5

71)

0.0

989

(16.0

93)*

0.9

053

(120.3

5)*

0.3

137

(0.2

23)

-0.0

135

(-0.8

06)

0.0

924

(-2.7

00)*

-5.5

46

-5.5

29

11,5

22.2

7

8.54

4

[0.74

1]

0.5

48

[0.7

339]


16/19

58


Table5:(cont.)Periodic-GARCH

(1,1)estimationofmeanequation(3)a

ndvarianceequation(12)

Coun

tries

0

1

1

2

s

1s

2s

AIC

SBC

LL

LBQ2(1

2)

ARCH-LM

(5)

INDON

ESIA

0.0

021

(3.9

23)*

0.0

008

(3.9

34)*

1.8

3E-06

(1.7

30)***

0.1

263

(18.0

10)*

0.8

826

(128.6

4)*

0.8

060

(5.8

32)*

-0.0

064

(-0.3

49)

0.1

388

(4.7

14)*

-5.4

61

-5.4

43

11,2

30.0

6

9.925

[0.623

]

0.8

81

[0.4

927]

ITALY

0.0

015

(3.8

48)*

0.0

002

(1.2

21)

2.5

2E-06

(4.5

35)*

0.1

010

(14.3

24)*

0.8

900

(101.6

1)*

0.2

931

(2.9

91)*

-0.0

445

(-2.8

42)

0.0

882

(2.2

25)**

-6.4

40

-6.4

19

11,9

71.4

5

15.57

8

[0.141

]

1.8

97

[0.1

090]

JAPAN

0.0

014

(4.4

63)*

0.0

005

(4.3

04)*

3.0

8E-06

(9.2

05)*

0.1

249

(28.8

08)*

0.8

663

(162.4

6)*

0.8

821

(3.0

98)*

0.0

002

(0.1

30)

0.0

343

(1.0

74)

-5.9

81

-5.9

72

21,8

23.8

7

4.553

[0.971

]

0.4

63

[0.8

036]

MALA

YSIA

0.0

011

(4.2

71)*

-2.2

8E-05

(-0.2

86)

5.4

2E-06

(3.0

37)*

0.1

864

(20.1

03)*

0.7

978

(80.3

44)*

0.0

983

(6.7

72)*

-0.0

821

(-3.4

62)*

0.2

029

(5.0

94)*

-6.2

25

-6.2

11

15,6

58.1

6

8.393

[0.754

]

0.5

42

[0.7

439]

MEX

ICO

0.0

023

(5.3

03)*

0.0

003

(1.4

47)

2.6

9E-05

(5.1

82)*

0.0

692

(3.4

76)*

0.9

182

(84.8

31)*

0.2

071

(3.7

17)*

-0.0

282

(-2.0

93)**

0.0

903

(2.9

74)*

-5.6

29

-5.6

17

16,0

08.4

3

15.94

5

[0.194

]

1.4

29

[0.2

102]

NETHER

LANDS

0.0

017

(5.6

30)*

0.0

002

(2.5

19)**

2.0

1E-06

(1.0

57)

0.1

060

(15.0

96)*

0.8

884

(106.3

7)*

-0.4

478

(-0.2

67)

-0.0

869

(-4.3

86)*

0.0

836

(1.9

44)***

-6.2

53

-6.2

41

16,7

38.3

6

18.69

3

[0.082

]

3.2

99

[0.0

056]

SINGA

PORE

0.0

015

(5.7

17)*

0.0

003

(1.6

54)***

6.1

1E-07

(0.4

51)

0.1

324

(20.2

12)*

0.8

485

(105.5

8)*

0.0

047

(0.0

07)

0.0

253

(1.2

89)

0.0

316

(0.9

34)

-5.7

35

-5.7

17

20,4

31.5

5

2.881

[0.992

]

0.2

80

[0.9

240]

TAIW

AN

0.0

027

(5.5

21)*

2.1

6E-05

(0.1

06)

1.9

1E-06

(3.2

21)*

0.0

640

(12.2

27)*

0.9

294

(129.1

9)*

-0.5

494

(-0.7

33)

0.0

527

(3.5

80)*

0.0

426

(1.2

29)

-5.5

78

-5.5

61

11,7

37.3

0

17.98

6

[0.116

]

2.6

30

[0.0

222]

UK-FTSE100

0.0

016

(6.7

60)*

9.9

7E-06

(3.3

04)*

1.0

4E-06

(2.4

88)**

0.0

958

(17.6

33)*

0.9

011

(123.0

8)*

0.0

332

(1.7

20)*

-0.0

670

(6.4

10)*

0.0

091

(0.2

66)

-6.5

66

-6.5

57

25,4

92.7

1

15.08

1

[0.237

]

1.9

35

[0.0

849]

US

S&P

0.0

009

(6.7

58)*

0.0

003

(6.9

07)*

6.2

6E-07

(4.7

80)*

0.0

834

(12.8

91)*

0.9

141

(148.8

9)*

0.7

155

(1.4

42)

-0.0

193

(-3.0

57)*

0.0

381

(4.6

58)*

-6.8

65

-6.8

62

54,7

89.8

6

15.63

6

[0.206

]

1.9

61

[0.1

809]

z-statisticbetweenbrackets,p-valuesbetweensquarebrackets,*denotessignificancein0.01level,**denotessignificancein0.05leveland***denotessignificancein0.10level

AIC

andSBCrefertoAkaikeandSchwarzinfo

rmationcriteria,LListheLogLikelihood,LBQ2istheLjung-Boxteston

squaredstandardizedresiduals


17/19

59


5. Conclusions

This study examined the turn of the month effect in 20 stock markets around the

globe using GARCH and PGARCH models. The results show that the turn of the month

effect is persistent in 19 out of 20 stock market indices during the whole period examined.Moreover, sub-sample periods have been explored too supporting the same concluding

remarks. In addition, when the post financial crisis period sample 2010-2013 is excluded

from the analysis, the turn of the month effect is present in all stock market indices.

The results of this study are consistent with earlier literature showing positive

returns at the beginning of the month and zero returns in the latter part of the month (see

e.g. Lakonishok and Smidt, 1988; Marquering et al., 2006). The paper provides several

important implications for investors and academic researchers. For investors this paper

gives useful information of the stock market behaviour during a calendar month and may

provide some ideas for profitable trading strategies. More specifically, the results establishedthat the stock market indices, examined in this study and regarding the turn of the month

effect, are not efficient, with the exception of Australia. Thus, investors can improve their

returns by timing their investment. However, given that the risks are also higher, extra

returns may not be obtainable.

Acknowledgements

The author would like to thank two anonymous referees for their valuable comments

and suggestions on the paper. Any remaining errors or omissions remain the responsibility

of the author.

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